Application of derivative to the study of the function

The monotony and continuity of functions

A sufficient condition for increasing functions

If at each point of the interval , the function is increasing on this interval

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A sufficient condition for decreasing functions

If at each point of the interval , the function is decreasing on this interval

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Remark. These conditions are only sufficient but not necessary conditions for growth and decreasing functions

A necessary and sufficient condition of constancy of the function

A function is constant on an interval if and only if when all points of the interval

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The extrema (maximums and minimums) of the function

The maximum point

Definition: the Point of defining functions is called a maximum point of this function if there is a neighborhood of the point that for all from this neighborhood inequality

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— the maximum point

— max

The point of a minimum

Definition: the Point of defining functions is called the minimum point of this function if there is a neighborhood of the point that for all from this neighborhood inequality

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— minimum point

— minimum

The critical point

Definition: the Interior points of the domain of definition of functions where the derivative is zero or does not exist are called critical

Necessary condition for an extremum

— the extremum point or not exists

(but not at each point where or not there will be an extremum!)

Sufficient condition of extremum

at the point of sign changes at point of maximum

at the point the sign changes at the point of minimum

An example of a graph of a function that has an extremum

— critical point

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The study of the function on the monotonicity and extrema

    Example.

  1. Find the domain of definition and the intervals on which the function is continuous
  2. Scope:

    The function is continuous at every point of its domain of definition

  3. To find the derivative
  4. Find the critical points, i.e. the interior points determining where or not there
  5. there is in the entire scope

    when

  6. Denote the critical point on the domain of definition, find sign of derivative and nature of function at each interval, which splits the definition area
  7. For each critical point determine whether it is high or low or is not an extremum point
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  9. Record porni the result of the study (intervals of monotonicity and extrema)
  10. increases with and

    subsides when

    Extreme points:

    Extremes:

Maximum and minimum values of continuous functions on the interval

Property: If the function is continuous on an interval and has therein a finite number of critical points, then it attains its maximum and minimum values on this interval either at a critical point belonging to this interval or at the endpoints of the interval

Finding maximum and minimum values of continuous functions on the interval

    Example. when

  1. To find the derivative
  2. Find the critical points ( or not exists)
  3. if and when

  4. Choose critical points that belong to a given segment
  5. A given segment belongs to only critical point

  6. To calculate the function values at critical points and the endpoints of the interval
  7. Compare the two values and choose the smallest and largest
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